Solving Fuzzy Integral Equations of the Second Kind by using the Reproducing Kernel Hilbert Space Method

Subject Areas : Statistics

Seddigheh  Farzaneh Javan ¹ (Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran)
Saeid  Abbasbandy ² (Department of Applied Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran)
Mohammad Ali  Fariborzi Araghi ³ (Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran)

Received: 2020-03-19 Accepted : 2020-08-24 Published : 2022-06-22

Keywords: معادلات انتگرال فازی نوع دوم, فضای هیلبرت هسته بازتولید, اعداد فازی, فرآیند گرام- اشمیت,

Abstract :

In this study, a new approach based on the Reproducing Kernel Hilbert Space Method is proposed to approximate the solution of the second kind fuzzy linear integral equations. For this purpose, at first by applying the concept of parametric form, the fuzzy integral equation is converted to a system of crisp integral equations. Then, this system is solved by using the reproducing kernel method free of the Gram-Schmidt orthogonalization process. Also, two numerical algorithms are proposed based on applying the Gram-Schmidt process and without using it. The general form of numerical solution accordingly the reproducing kernel method is introduced and the convergence theorem of solution of the proposed scheme to the exact solution is proved. Finally, a sample fuzzy integral equation is solved by means of both suggested algorithms and the results are compared for differents points and levels. Due to the difficulties in applying the Gram-Schmidt process, the obtained results of the new algorithm are satisfactory.

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